Nonlinear Hodge correspondence for morphisms

We study Higgs sections, flat sections, and their higher-dimensional and functorial analogues in the setting of nonlinear harmonic bundles. For a harmonic vector bundle over a compact K\"ahler manifold, a global section is flat if and only if it is a Higgs section. We show that for general nonlinear harmonic bundles this equivalence requires the vanishing of a degree obstruction. We then extend the result to sub-fibrations and to morphisms by interpreting morphisms as graph sub-fibrations in a fiber product.